For questions on continued fractions.
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Is the Iterated Continued fraction from Convergent​s for Pi/2 exactly 3/2?
Iterated continued fraction from convergents are described at https://oeis.org/wiki/Convergents_constant and https://oeis.org/wiki/Table_of_convergents_constants.
Do you think there is any error in ...
-1
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3answers
1k views
Extract a Pattern of Iterated continued fractions from convergents
I have been working on an article at
https://oeis.org/wiki/Table_of_convergents_constants
where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) ...
7
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1answer
213 views
What causes the convergence of Iterated continued fractions from convergents?
Here is a small discovery I stumbled across a few weeks ago. I hope at least one person will find it interesting enough to help me.
The iterated continued fractions from convergents (or convergents ...
15
votes
2answers
708 views
Motivation behind this eccentric Ramanujan Identity
I just visited the MathJaX page due to the Math.SE website showing some problems while loading the page. I saw some demo math equations samples at this page, when this identity actually caught my ...
4
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0answers
198 views
How to find the number of continued fraction from a periodic representation?
Problem
Find the number that represented by $[2,2,2 \ldots]$
I know it wasn't difficult, but I was absent the last two classes. So I just want to make sure that I got it right.
My attempt was,
...
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1answer
128 views
How to find continued fraction of the form $a\sqrt{b}$?
For the form $\sqrt{b}$, I could just apply the recursive quadratic formula:
$$P_{k+1} = a_kQ_k - P_k$$
$$Q_{k+1} = \dfrac{d - P^2_{k+1}}{Q_k}$$
$$\alpha_k = \dfrac{P_k + \sqrt{d}}{Q_k}$$
...
11
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1answer
322 views
Deriving a trivial continued fraction for the exponential
Lately, I learned about the following continued fraction for the exponential function:
$$\exp(x)=1+\cfrac{x}{1-\cfrac{x/2}{1+x/2-\cfrac{x/3}{1+x/3-\cfrac{x/4}{1+x/4-\dots}}}}$$
I thought it was ...
6
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2answers
188 views
A question on continued fraction
Let $a$ be a positive irrational number. Let $p_k/q_k, p_{k+1}/q_{k+1}$ be two consecutive
convergents of its simple continued fraction, where $k\ge 1$.
Is it possible that both ...
10
votes
3answers
473 views
Proving the continued fraction representation of $\sqrt{2}$
There's a question in Spivak's Calculus (I don't happen to have the question number in front of me in the 2nd Edition, it's Chapter 21, Problem 7) that develops the concept of continued fraction, ...
2
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1answer
230 views
Solving equations using continued fractions?
We solve the pell equation using the continued fraction for square root of 2. What equations can we solve using the continued fraction of cube roots (and other numbers too)?
6
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0answers
387 views
What can Euler's identity teach us about (generalised) continued fractions?
We know that $$e^{i \pi} = -1 .$$ We can transform all of the components of this identity into (generalized) continued fractions. When we start of with $\pi$, we see that $$ \Big(3+ ...
5
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1answer
213 views
What is the length of a continued fraction expansion of a rational number?
I was reviewing quantum factorization and am slightly unclear on a classical detail of order-finding.
Given a (suitably nice) periodic function $f$ with unknown period $r$ and a power of two $N > ...
10
votes
1answer
381 views
Continued Fraction expansion of tan(1)
Prove that continued fraction of tan(1)=[1;1,1,3,1,5,1,7,1,9,1,11,...]. I tried using the same sort of trick used for finding continued fractions of quadratic irrationals and trying to find a ...
4
votes
2answers
251 views
Adding integers to an infinite continued fraction expansion doesn't change the value?
I'm learning about continued fractions, and I've enjoyed them so far, but I'm unsure if I've done the following correctly. I have no real experience with analysis, so I'm not sure if my reasoning is ...
5
votes
1answer
348 views
Continued Fraction of an Infinite Sum
What is the continued fraction for $\displaystyle\sum_{i=1}^n\frac{1}{2^{2^i}}$
It seems to be "almost" periodic, but I can't figure out the exact way to express it.
6
votes
3answers
571 views
Why are some mathematical constants irrational by their continued fraction while others aren't?
Catalan's Constant and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On wikipedia (I'm sorry, I can't post anymore ...
10
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2answers
829 views
How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized continued fraction?
Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the Gauss Notation for generalized continued fractions.
Description of the Bauer-Muir transformation
...
11
votes
2answers
666 views
Continued fraction for $\frac{1}{e-2}$
A couple of years ago I found the following continued fraction for $\frac1{e-2}$:
$$\frac{1}{e-2} = 1+\cfrac1{2 + \cfrac2{3 + \cfrac3{4 + \cfrac4{5 + \cfrac5{6 + \cfrac6{7 + \cfrac7{\cdots}}}}}}}$$
...
4
votes
1answer
573 views
A nicer proof of Lagrange's 'best approximations' law?
Let $p_N/q_N$ be the $N^\text{th}$ convergent of the continued fraction for some irrational number $\alpha$. It turns out that for any other approximation $p/q$ (with $q \le q_N$) which isn't a ...
